ON THE CHOLESKY FACTORIZATION OF THE GRAM MATRIX OF LOCALLY SUPPORTED FUNCTIONS

Cholesky factorization of bi-infinite and semi-infinite matrices is studied and in particular the following is proved. If a bi-infinite matrix A has a Cholesky factorization whose lower triangular factor L and its lower triangular inverse decay exponentially away from the diagonal, then the semi-infinite truncation of A has a lower triangular Cholesky factor whose elements approach those of L exponentially. This result is then applied to studying the asymptotic behavior of splines obtained by orthogonalizing a large finite set of B-splines, in particular identifying the limiting profile when the knots are equally spaced.